Mathematical Strategies for Soccer: Pt.1 Team Formations

Game theory and decision theory are powerful and interconnected pillars of strategic thinking. Both theories are useful tools that can create an advantage in competitive sports, particularly soccer.

Decision theory focuses on the process by which an individual makes the best possible decision given their understanding of the world. It is a systematic way to view all possible outcomes of a decision, map out scenarios, and predict outcomes.

I.e., a soccer player's decisions on the field, such as whether to pass or shoot, are based on their perception of the game situation.

Game theory is the study of strategic interaction. It centers around the decisions of multiple players, with each player trying to make the best decision based on their assumptions about the others' choices.

This newsletter is the first edition in a three-part series that explores how the application of game theory can offer a competitive edge to coaches by providing unique and strategic perspectives.

This series underscores the value of game theory not only as an academic concept but as a practical tool for gaining advantages on the field.

The soccer-specific areas of focus will be:

1. Team formations

2. Penalty Kicks

3. Time wasting

Team Formations

One of the most useful applications of game theory in soccer is the determination of team formations. The choice between a 4-4-2, 4-3-3, 4-5-1, or any other formation is a strategic response to the anticipated strengths and weaknesses of the opposition. The formation is designed to maximize the team's strengths while exploiting the opponent's vulnerabilities.

A Closer Look

Assume Perfect Information

Let's insert a few conditions:

Teams: Team Alpha vs Team Beta

1. Weather: It's raining heavily on the day of the match, which makes it harder for teams to play a controlled, offensive game.
2. Player Fatigue: Team Alpha has had a congested schedule, and their key offensive players are tired, which could hamper their attacking potency.
3. Injuries: Team Beta's key defender is injured and can't participate in the match.

These conditions could modify our payoff matrix as follows:

• If both teams play defensively, the conditions (weather, fatigue, and injuries) are likely to result in a low-scoring game, perhaps even without any goals (0,0).
• If Team Alpha goes offensive (4-3-3) and Team Beta stays defensive (4-5-1), given the lack of Team Beta's key defender and the weather conditions hampering a more controlled offensive play, Team Alpha may score 1 goal and Team Beta may not score at all (1,0).
• If Team Alpha stays defensive (5-4-1) and Team Beta goes offensive (4-3-3), the fatigue of Team Alpha's players and the weather may result in a draw with each team scoring 1 goal (1,1).
• If both teams play offensively (4-3-3), despite the weather conditions and player fatigue, the absence of Team Beta's key defender could lead to a high-scoring draw, with each team scoring 2 goals (2,2).

Now, Team Alpha might be more inclined to play offensively, even with tired key attackers, because they recognize Team Beta's vulnerability in defense due to the injury. Similarly, Team Beta might choose a defensive strategy, recognizing the potential risk of their weakened defense and the effect of the weather on offensive play.

There are countless other factors that could be considered, but this illustrates how additional variables can be integrated into the game theory analysis.

Lets Assume: Imperfect information

In a situation with limited information, teams need to make decisions based on their expectations and the information that is available to them. They might also use strategies to try to uncover more information or mislead the other team.

Here's an example:

Assume that both teams have limited information about the other team's formation decision. However, they know each other's general strengths and weaknesses. Everyone knows that Team Alpha has a strong defense and Team Beta has a strong offense.

Let's also assume that Team Alpha doesn't know about the injury to Team Beta's key defender, and Team Beta doesn't know about the fatigue of Team Alpha's key offensive players.

Given this limited information, the teams might make the following strategic decisions:

• Team Alpha, knowing their own strength in defense and not knowing about the injury in Team Beta's defense, might choose a defensive formation (5-4-1) expecting to hold off Team Beta's strong offense and hoping to score on a counter-attack.
• Team Beta, knowing their strength in offense and not knowing about the fatigue of Team Alpha's key offensive players, might choose an offensive formation (4-3-3), hoping to overcome Team Alpha's strong defense.

This decision-making process is more about risk management and expectations than definitive outcomes. The teams are making the best decisions they can based on the limited information they have. This adds a layer of complexity to the game and makes strategic decisions even more crucial.

Once the game starts and the teams see each other's formations, they can adjust their strategies accordingly.

This is where the skill of the coach in making tactical decisions comes in. There are many other sources of uncertainty and many other factors that teams would need to consider. But this example illustrates the basic idea of how teams might approach decision-making under conditions of limited information.